From original vinyl to t-shirts and merchandise, it’s hard to think of a more iconic image in the music industry. Yet surprisingly, there’s a lot of math behind the simple geometric design and rainbow-patterned illustrations.
As reported by Chalkdust Magazinethe album cover shows a beam of white light hitting a triangular prism splitting into “constituent parts” (which form the rainbow).
This is an example of an “optical phenomenon” known as refraction and scattering. This explains why the rainbow and the incoming beam are not parallel and how the light splits as it passes through the prism.
Sean Jamshidi, a PhD student at University College London, broke down three important mathematical elements from the album cover.
The refraction inside the album cover is related to the speed at which a beam of light travels (depending on the medium) which can be glass, air or water.
Given the medium, mathematics can calculate the index of refraction:
Briefly, c is the speed of light in vacuum and v is the speed of light through the medium.
n is close to 1, whereas for glass it is around 1.5 – so light travels slower through glass than through air.
Light normally takes the fastest path and the difference in speed between air and glass means that the shortest path in terms of distance may not be the fastest.
Moving on to angles, θ1 and θ2 are measured from the normal to the material interface.
As glass has a higher refractive index than air (n2>n1n2>n1), a light beam passing from air into glass will bend towards the normal (θ1>θ2θ1>θ2).
When the beam leaves the other side of the prism, the reverse happens and the angle increases again.
In terms of the angle change on the prism, it can be calculated by a quadrilateral (composed of the prism and the beam of light.)
It is an example of “Snell’s Law” which in mathematics is described as “a formula used to describe the relationship between the angles of incidence and refraction”.
Sean said: “Given an entry angle θ1, Snell’s law allows us to calculate θ2. We can then add the angles in the red quadrilateral, which gives θ3.
“In terms of θ1 and the refractive indices n1 and n2, we again apply Snell’s law at the point where the light exits to give us θ4.”
The multicolored beam is the only glimpse of color we get on this album cover, and while the rainbow is minimal, it contrasts against the black background.
Sean explained: “Where λ is the wavelength of light, A and b are constants that depend on the material.
“In air, the value of b is very small (about 10−18) and therefore the index of refraction can be set to a constant, which means that our light beam arrives at the prism with all colors striking at the same place and at the same time.”
Designer Hipgnosis knew a thing or two about math when designing the album cover. By combining the concepts of refraction and dispersion, we can calculate the exact path of the light beam.
Next time you listen to Pink Floyd, take a look at the album cover and prepare to have your mind blown by the math in front of you.